Questions: Given the function g(n)=(n-2)(n+6)(n-7) : the coordinates of its g-intercept are the coordinates of its n-intercepts are

Transcript text: Given the function $g(n)=(n-2)(n+6)(n-7)$ : the coordinates of its $g$-intercept are $\square$ the coordinates of its $n$-intercepts are $\square$

Solution

Solution Steps

Solution Approach

To find the coordinates of the $ g $-intercept of the function $ g(n) = (n-2)(n+6)(n-7) $, we need to evaluate the function at $ n = 0 $. This will give us the $ g $-intercept, which is the point where the graph of the function crosses the $ g $-axis. For the $ n $-intercepts, we need to find the values of $ n $ that make the function equal to zero. These are the roots of the polynomial, which can be found by setting each factor equal to zero.

Step 1: Calculate the $ g $-intercept

To find the $ g $-intercept, we evaluate the function at $ n = 0 $: \[ g(0) = (0 - 2)(0 + 6)(0 - 7) = (-2)(6)(-7) = 84 \] Thus, the coordinates of the $ g $-intercept are $ (0, 84) $.

Step 2: Find the $ n $-intercepts

The $ n $-intercepts occur where the function equals zero: \[ g(n) = (n - 2)(n + 6)(n - 7) = 0 \] Setting each factor to zero gives us the solutions: